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I have to find the value of the integral $$\int_{0}^{\frac{\pi}{2}} \frac{\sin^2(x)}{\sin(x)+\cos(x)}$$ I was able to can write it as $$\frac{1}{\sqrt{2}}\int_0^\frac{\pi}{2}\frac{\sin^2(x)}{\sin(x+\frac{\pi}{4}) }$$

But I cannot go any further.

Edit: Also I am looking for a short method

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  • $\begingroup$ Following Lab's answer below (which is a quick approach), you can find the anti derivative of $\frac{1}{sinx+cosx}$ through tangent half angle substitution OR use a standard anti derivative for $cscx$ where the $x$ is read as $x+\pi/4$ $\endgroup$
    – imranfat
    Mar 16, 2016 at 16:43

1 Answer 1

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HINT:

Using $\int_a^bf(x)\ dx=\int_a^bf(a+b-x)\ dx,$

$$I+I=\dfrac1{\sqrt2}\int_0^{\pi/2}\dfrac{dx}{\sin\left(x+\dfrac\pi4\right)}$$

Hope you can take it from here?

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    $\begingroup$ I think your speed of typing is faster than my speed of thinking...:) $\endgroup$
    – imranfat
    Mar 16, 2016 at 16:44
  • $\begingroup$ Yes. Got it. Thanks $\endgroup$
    – Sayan Chattopadhyay
    Mar 16, 2016 at 16:47

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